Question

A random sample of n = 45 observations from a quantitative population produced a mean x = 2.5 and a standard deviation s = 0.26. Your research objective is to show that the population mean μ exceeds 2.4. Calculate the p-value for the test statistic z = 2.58. (Round your answer to four decimal places.)

Answer 1

P-value (t=2.58) = 0.0066.

Note: as we are using the sample standard deviation, a t-statistic is appropiate instead os a z-statistic.

As the P-value (0.0066) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the population mean μ exceeds 2.4.

Explanation:

This is a hypothesis test for the population mean.

The claim is that the population mean μ exceeds 2.4.

Then, the null and alternative hypothesis are:

`H_0: \mu=2.4\\\\H_a:\mu> 2.4`

The significance level is 0.05.

The sample has a size n=45.

The sample mean is M=2.5.

As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=0.26.

The estimated standard error of the mean is computed using the formula:

`s_M=(s)/(√(n))=(0.26)/(√(45))=0.0388`

Then, we can calculate the t-statistic as:

`t=(M-\mu)/(s/√(n))=(2.5-2.4)/(0.0388)=(0.1)/(0.0388)=2.58`

The degrees of freedom for this sample size are:

`df=n-1=45-1=44`

This test is a right-tailed test, with 44 degrees of freedom and t=2.58, so the P-value for this test is calculated as (using a t-table):

`P-value=P(t>2.5801)=0.0066`

As the P-value (0.0066) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the population mean μ exceeds 2.4.